1. Problem: Ana starts with 8 and counts by 5s. Her first three numbers are 8, 13, and 18. Josh starts with a whole number other than 8 and counts by a whole number other than 5. Some of his numbers are 11, 32, and 46, but he does not start with 11. 24 is not one of his numbers. What number does Josh start with? 2. Model: Let Josh's starting number be $s$ and his common difference be $d$, where $s$ and $d$ are whole numbers with $s\ne8$ and $d\ne5$. 3. Because 11, 32, and 46 are in his list there exist nonnegative integers $a,b,c$ with $s+ad=11$, $s+bd=32$, and $s+cd=46$. 4. Take differences: $32-11=21$, $46-32=14$, and $46-11=35$. 5. Conclusion on $d$: Since $d$ divides each of these differences, we have $d\mid21$, $d\mid14$, and $d\mid35$, so $d\mid\gcd(21,14,35)=7$. 6. Possibilities: Thus $d=1$ or $d=7$. 7. Eliminate $d=1$: If $d=1$ the sequence would include every integer from some starting point; because 11 is included that would force 24 to be included, contradicting the given that 24 is not one of his numbers. Therefore $d=7$. 8. Simplify (explicit cancellation example): $$\frac{21}{7}=\frac{\cancel{7}\cdot3}{\cancel{7}}=3$$ 9. Solve for $s$ modulo $d$: From $s+ad=11$ with $d=7$ we get $s\equiv11\pmod7$, so $s\equiv4\pmod7$ and hence $s=4+7k$ for some integer $k$. 10. Determine the start: Because 11 appears in the list and Josh does not start with 11, the starting number must be at or before 11 in the progression. The nonnegative residue congruent to 4 and less than 11 is $s=4$. 11. Check: Starting at $4$ and counting by $7$ gives $$4,11,18,25,32,39,46$$ which contains 11, 32, and 46, does not contain 24, and the start is not 8, so all conditions are met. 12. Answer: Josh starts with $4$.